L(s) = 1 | + (0.806 + 1.16i)2-s − i·3-s + (−0.699 + 1.87i)4-s + (1.16 − 0.806i)6-s + 0.746·7-s + (−2.74 + 0.699i)8-s − 9-s + 5.36i·11-s + (1.87 + 0.699i)12-s + 2.92i·13-s + (0.601 + 0.866i)14-s + (−3.02 − 2.62i)16-s + 2.13·17-s + (−0.806 − 1.16i)18-s + 1.73i·19-s + ⋯ |
L(s) = 1 | + (0.570 + 0.821i)2-s − 0.577i·3-s + (−0.349 + 0.936i)4-s + (0.474 − 0.329i)6-s + 0.282·7-s + (−0.968 + 0.247i)8-s − 0.333·9-s + 1.61i·11-s + (0.540 + 0.201i)12-s + 0.811i·13-s + (0.160 + 0.231i)14-s + (−0.755 − 0.655i)16-s + 0.517·17-s + (−0.190 − 0.273i)18-s + 0.397i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.09650 + 1.41132i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.09650 + 1.41132i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.806 - 1.16i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 0.746T + 7T^{2} \) |
| 11 | \( 1 - 5.36iT - 11T^{2} \) |
| 13 | \( 1 - 2.92iT - 13T^{2} \) |
| 17 | \( 1 - 2.13T + 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 - 7.49T + 23T^{2} \) |
| 29 | \( 1 - 6.74iT - 29T^{2} \) |
| 31 | \( 1 - 2.64T + 31T^{2} \) |
| 37 | \( 1 + 1.07iT - 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 + 7.44iT - 43T^{2} \) |
| 47 | \( 1 - 1.73T + 47T^{2} \) |
| 53 | \( 1 + 7.72iT - 53T^{2} \) |
| 59 | \( 1 - 6.85iT - 59T^{2} \) |
| 61 | \( 1 + 6.45iT - 61T^{2} \) |
| 67 | \( 1 + 7.44iT - 67T^{2} \) |
| 71 | \( 1 - 13.2T + 71T^{2} \) |
| 73 | \( 1 + 0.690T + 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 - 5.85iT - 83T^{2} \) |
| 89 | \( 1 - 7.59T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.15439272160168492547802129855, −9.856593959508064893103253404790, −8.937839683200887053477301436268, −8.049172424857950598677491643532, −7.02205000635388326457640921666, −6.78299399252938054796200938983, −5.32946146228741333314779173856, −4.66382880803750687960697947190, −3.36937023508344912732079935475, −1.88015672129122475097908214960,
0.881974301390377661719713125968, 2.82148910602230004932074902259, 3.50054771203984655960273502930, 4.77413096614882401058908058219, 5.52931135472097781278342657109, 6.42806484619486163230968327003, 8.051358120419233909427017440717, 8.841572055251705857021683828748, 9.773013455571653581663699652960, 10.61707880311331032738830003985